Reduced homology and cohomology theories

Definition 3.1.2 (Homology and cohomology theories). A reduced homology theory is a pair \((\widetilde E_*, \sigma )\) consisting of a functor \[ \widetilde E_*\colon \An _* \to \Ab ^{\Z } \] and a natural isomorphism \[ \sigma \colon \widetilde E_*(-) \iso \widetilde E_{*+1}(\Sigma (-)), \] called the suspension isomorphism, satisfying:

(1)

(Wedge axiom) For every collection \((X_i)_{i \in I}\) of pointed animae, the canonical map \[ \bigoplus _{i \in I}\widetilde E_*(X_i) \to \widetilde E_*\Big (\bigvee _{i \in I}X_i\Big ) \] is an isomorphism.

(2)

(Exactness) For every cofiber sequence \(X \xrightarrow {f} Y \xrightarrow {g} Z\) in \(\An _*\), the sequence of abelian groups \[ \widetilde E_*(X) \xrightarrow {f_*} \widetilde E_*(Y) \xrightarrow {g_*} \widetilde E_*(Z) \] is exact in each degree.

Dually, a reduced cohomology theory is a pair \((\widetilde E^*, \sigma )\) consisting of a functor \(\widetilde E^*\colon \An _*\catop \to \Ab ^{\Z }\) and a natural isomorphism \(\sigma \colon \widetilde E^*(-) \iso \widetilde E^{*+1}(\Sigma (-))\) satisfying:

(1)

(Wedge axiom) For every collection \((X_i)_{i \in I}\) of pointed animae, the canonical map \(\widetilde E^*\Big (\bigvee _{i \in I}X_i\Big ) \to \prod _{i \in I}\widetilde E^*(X_i)\) is an isomorphism.

(2)

(Exactness) For every cofiber sequence \(X \xrightarrow {f} Y \xrightarrow {g} Z\) in \(\An _*\), the sequence \(\widetilde E^*(Z) \xrightarrow {g^*} \widetilde E^*(Y) \xrightarrow {f^*} \widetilde E^*(X)\) is exact in each degree.